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Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"ad71da399ad20cedafda30f804d07f63755db3be67adf69ce669a017fd162c17","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-abf94b4038449932","role":"sequence","name":"veronese_section_2_1 of Cube of lower triangular normalized binomial matrix.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:     0;\n  1:     1;\n  2:     6;\n  3:    27,      4;\n  4:   108,     48;\n  5:   405,    360,     9;\n  6:  1458,   2160,   162;\n  7:  5103,  11340,  1701,   16;\n  8: 17496,  54432, 13608,  384;\n  9: 59049, 244944, 91854, 5184, 25;\n","recurrence_sentence":"The row polynomials satisfy (972*n-972-(270*n^3-756*n^2+594*n-108)*x+(3*n^5-9*n^4+9*n^3-3*n^2)*x^2)*P(n,x) = (2916*n-(810*n^3-1458*n^2-972*n+2592)*x+(9*n^5-18*n^4-153*n^3+306*n^2+108*n-144)*x^2+(n^5-3*n^3-2*n^2)*x^3)*P(n-1,x) + (5832*x-(1620*n^2+324*n)*x^2+(18*n^4+36*n^3+198*n^2+36*n-72)*x^3-(2*n^4+4*n^3+2*n^2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((2916*n - (810*n^3 - 1458*n^2 - 972*n + 2592)*x + (9*n^5 - 18*n^4 - 153*n^3 + 306*n^2 + 108*n - 144)*x^2 + (n^5 - 3*n^3 - 2*n^2)*x^3)*P[n - 1, x] + (5832*x - (1620*n^2 + 324*n)*x^2 + (18*n^4 + 36*n^3 + 198*n^2 + 36*n - 72)*x^3 - (2*n^4 + 4*n^3 + 2*n^2)*x^4)*D[P[n - 1, x], x])/(972*n - 972 - (270*n^3 - 756*n^2 + 594*n - 108)*x + (3*n^5 - 9*n^4 + 9*n^3 - 3*n^2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((2916*n - (810*n^3 - 1458*n^2 - 972*n + 2592)*x + (9*n^5 - 18*n^4 - 153*n^3 + 306*n^2 + 108*n - 144)*x^2 + (n^5 - 3*n^3 - 2*n^2)*x^3)*v[i - 1] + (5832*x - (1620*n^2 + 324*n)*x^2 + (18*n^4 + 36*n^3 + 198*n^2 + 36*n - 72)*x^3 - (2*n^4 + 4*n^3 + 2*n^2)*x^4)*deriv(v[i - 1], x))/(972*n - 972 - (270*n^3 - 756*n^2 + 594*n - 108)*x + (3*n^5 - 9*n^4 + 9*n^3 - 3*n^2)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-8d887bfc40d036b9","steps":["Veronese 2 (j=1)","(\u03b8+1)\u00b2"],"edges":["e-59cb5e4e13a9546c","e-12dfb86592e8bb4d"],"text":"A027465 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2"},"headline":"A027465 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2","chain":null}